2/
One might say that physicists study the symmetry of nature, while mathematicians study the nature of symmetry.
1/
2/
3/
1) Closure: a⊡b is in G.
2) Associativity: (a⊡b)⊡c = a⊡(b⊡c)
3) Identity: e⊡a = a⊡e = a.
4) Inverse: There exists an element a* such that a*⊡a = a⊡a* = e.
4/
5/
then μ: G ⟼ H is a "group morphism" if for all elements of G:
μ(a⊡b) = μ(a)⊠μ(b). Note that for all a:
μ(a)⊠ε = μ(a) = μ(a⊡e) = μ(a)⊠μ(e)
and hence μ(e)=ε; Similarly it can be shown that
μ(a*) = μ(a)*.
- the *Baby Monster*, *B*, of size
2⁴¹ ⋅ 3¹³ ⋅ 5⁶ ⋅ 7² ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 31 ⋅ 47; and
2⁴⁶ ⋅ 3²⁰ ⋅ 5⁹ ⋅ 7⁶ ⋅ 11² ⋅ 13³ ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71.
Counting up the number of distinct primes in that last number gives us 15.
The number of distinct prime factors
in the size, n, of the *Monster Group* M
is
15.
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